ON INTEGER SETS WITH THE SAME REPRESENTATION FUNCTIONS
| Autor: | KAI-JIE JIAO, CSABA SÁNDOR, QUAN-HUI YANG, JUN-YU ZHOU |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Bulletin of the Australian Mathematical Society. 106:224-235 |
| ISSN: | 1755-1633 0004-9727 |
| Popis: | Let $\mathbb {N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb {N}$ and $n\in \mathbb {N}$ , let $R_S(n)$ denote the number of solutions of the equation $n=s_1+s_2$ , $s_1,s_2\in S$ and $s_1. Let A be the set of all nonnegative integers which contain an even number of digits $1$ in their binary representations and $B=\mathbb {N}\setminus A$ . Put $A_l=A\cap [0,2^l-1]$ and $B_l=B\cap [0,2^l-1]$ . We prove that if $C \cup D=[0, m]\setminus \{r\}$ with $0, $C \cap D=\emptyset $ and $0 \in C$ , then $R_{C}(n)=R_{D}(n)$ for any nonnegative integer n if and only if there exists an integer $l \geq 1$ such that $m=2^{l}$ , $r=2^{l-1}$ , $C=A_{l-1} \cup (2^{l-1}+1+B_{l-1})$ and $D=B_{l-1} \cup (2^{l-1}+1+A_{l-1})$ . Kiss and Sándor [‘Partitions of the set of nonnegative integers with the same representation functions’, Discrete Math.340 (2017), 1154–1161] proved an analogous result when $C\cup D=[0,m]$ , $0\in C$ and $C\cap D=\{r\}$ . |
| Databáze: | OpenAIRE |
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